#6343144

order	512
nilpotency class	3
isoclinism	isoclinic class of #61569
automorphism	Aut(G) order 262144
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	C₂×#3483	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,9)(4,13)(5,10)(6,15)(7,11)(8,12)(14,16)(17,18,21,24,22,26,20,25)(19,27,29,23,30,31,28,32);(1,3,7,4)(2,8,11,16)(5,12,6,14)(9,10,13,15)(17,19,22,30)(18,23,26,32)(20,28,21,29)(24,31,25,27);(1,4,7,3)(2,9,11,13)(5,12,6,14)(8,10,16,15)(17,20,22,21)(18,24,26,25)(19,28,30,29)(23,31,32,27);(1,5,7,6)(2,10,11,15)(3,12,4,14)(8,13,16,9)(17,20,22,21)(18,25,26,24)(19,28,30,29)(23,27,32,31);(2,11)(8,16)(9,13)(10,15)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,e^a=ehi,f^a=fh,f^b=fh,d^c=dhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250340);